Question

Draw the graph of the equations x + 2y = 3, 2x + y = 3 and x – y = 0. Show that these three lines pass through the same point. Find the co-ordinates of this common point.

Answer 1

1. Solving the equations algebraically

The given equations are:

1. x + 2y = 3
2. 2x + y = 3
3. x – y = 0

Step 1: Solve Equation (3)

From equation (3),

x – y = 0, we get x = y.

Step 2: Substitute x = y into equation (2)

Substituting x = y into 2x + y = 3:

2(y) + y = 3

3y = 3

y = 1

Since x = y, we have x = 1.

Step 3: Verify with equation (1)

Substitute x = 1 and y = 1 into x + 2y = 3:

1 + 2(1) = 3

3 = 3

Since the values satisfy all three equations, the common point of intersection is (1, 1).

2. Graphical representation

To draw the graph, we find at least two points for each line:

For x + 2y = 3:

If x = 1, y = 1; if x = 3, y = 0.

For 2x + y = 3:

If x = 1, y = 1; if x = 0, y = 3.

For x – y = 0:

If x = 0, y = 0; if x = 1, y = 1.

The three lines pass through the same point because the coordinates (1, 1) satisfy all three equations simultaneously. The coordinates of the common point are (1, 1).